Who pays to restore the heat
No. In California, providing heat and maintaining the heating system are the landlord's responsibility, so you generally should not have to pay to repair a furnace or heater — unless you clearly caused the damage.
The cost of fixing or replacing a failed furnace or heater falls on the landlord as part of keeping the rental livable. A tenant might be responsible only if they damaged the equipment through clear misuse, which is uncommon with a heating system.
If a landlord tries to charge you for heating repairs that are really their duty, or expects you to supply your own heat, that is worth questioning before you agree or pay anything. Keep any communications about who is responsible. If your landlord is pushing heating-repair costs onto you, a free review can help you understand what is fair.
This is general information about California tenant rights, not legal advice. Every situation is different.